Download Using a graphic display calculator

TI-84 Plus
Using a graphic display
calculator
17
CHAPTER OBJECTIVES:
This chapter shows you how to use your graphic display calculator
(GDC) to solve the different types of problems that you will meet in your
course. You should not work through the whole of the chapter – it is simply
here for reference purposes. When you are working on problems in the
mathematical chapters, you can refer to this chapter for extra help with
your GDC if you need it.
Instructions for the TI-84 Plus calculator
Use this list to help you to find the topic you need
Chapter contents
1 Functions
1.1
Graphing linear functions
Finding information about the graph
1.2
Finding a zero
1.3
Finding the gradient (slope) of a line
1.4
Solving simultaneous equations
graphically
Simultaneous and quadratic equations
1.5
Solving simultaneous linear equations
Quadratic functions
1.6
Drawing a quadratic graph
1.7
Solving quadratic equations
1.8
Finding a local minimum or maximum
point
Exponential functions
1.9
Drawing an exponential graph
1.10 Finding a horizontal asymptote
Logarithmic functions
1.11 Evaluating logarithms
1.12 Finding an inverse function
1.13 Drawing a logarithmic graph
Trigonometric functions
1.14 Degrees and radians
1.15 Drawing trigonometric graph
More complicated functions
1.16 Solving a combined quadratic and
exponential equation
1.17 Using sinusoidal regression
3 Integral calculus
3.1
Finding the value of an indefinite
2
2
3
4
6
7
7
8
12
13
14
14
15
16
17
17
19
2 Differential calculus
Finding gradients, tangent and maximum and
minimum points
2.1
Finding the gradient at a point
20
2.2 Drawing a tangent to a curve
21
2.3 Finding maximum and minimum
2.4
2.5
points
Finding a numerical derivative
Graphing a numerical derivative
21
23
23
integral
Finding the area under a curve
3.2
4 Vectors
Scalar product
4.1
Calculating a scalar product
4.2 Calculating the angle between two
vectors
5.5
5.6
28
29
5 Statistics and probability
Entering data
5.1
Entering lists of data
5.2 Entering data from a frequency table
Drawing charts
5.3 Drawing a frequency histogram from
5.4
26
26
a list
Drawing a frequency histogram from a
frequency table
Drawing a box and whisker diagram
from a list
Drawing a box and whisker diagram from
a frequency table
30
31
31
32
32
33
Calculating statistics
5.7 Calculating statistics from a list
5.8 Calculating statistics from a frequency
34
table
5.9 Calculating the interquartile range
5.10 Using statistics
35
35
36
Calculating binomial probabilities
5.11 Use of nCr
5.12 Calculating binomial probabilities
Calculating normal probabilities
5.13 Calculating normal probabilities from
X-values
5.14 Calculating X-values from normal
probabilities
36
37
39
40
Scatter diagrams, linear regression and the
correlation coefficient
5.15 Scatter diagrams
41
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Using a graphic display calculator
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TI-84 Plus
Before you start
You should be familiar with:
●
●
●
●
●
Important keys on the keyboard: ON 2nd DEL CLEAR
The home screen
Changing window settings in the graph screen
Using zoom tools in the graph screen
Using trace in the graph screen
Y= X, T, H, n ENTER GRAPH
For a reminder of how to
perform the basic operations
have a look at your GDC
manual.
1 Functions
1.1 Graphing linear functions
Example 1
Draw the graph of the function y = 2x + 1.
Press Y= to display the Y= editor. The default graph type is Function,
so the form Y= is displayed.
Type 2x + 1 and press ENTER .
Press ZOOM | 6:ZStandard to use the default axes which are −10 ≤ x ≤ 10
and −10 ≤ y ≤ 10.
The graph of y = 2x + 1 is now displayed on the screen.
Finding information about the graph
The GDC can give you a lot of information about the graph of a function,
such as the coordinates of points of interest and the gradient (slope).
1.2 Finding a zero
The x-intercept is known as a zero of the function.
Example 2
Find the zero of y = 2x + 1.
Draw the graph of y = 2x + 1 as in Example 15.
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TI-84 Plus
Press
2nd CALC
Press
ENTER .
| 2:Zero.
To find the zero you need to give the left and right bounds of a region that
includes the zero.
The calculator shows a point and asks you to set the left bound.
Move the point using the
and
keys to choose a position to
the left of the zero.
Press ENTER .
The calculator shows another point and asks you to set the right bound.
and
keys so that the region
Move the point using the
between the left and right bounds contains the zero.
When the region contains the zero press
ENTER .
Press ENTER again to supply a guess for the value of the zero.
The calculator displays the zero of the function y = 2x + 1 at the
point (−0.5, 0).
1.3 Finding the gradient (slope) of a line
The correct mathematical notation for gradient (slope) is
dy
. You will find
dx
out more about this in the chapter on differential calculus. Here we just need to
know this is the notation that will give us the gradient (slope) of the line.
Example 3
Find the gradient of y = 2x + 1.
First draw the graph of y = 2x + 1 (see Example 15).
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Press
2nd CALC
Press
ENTER .
| 6: dy/dx.
Select any point on the line using the
and
keys and press
ENTER .
The gradient (slope) is 2.
1.4 Solving simultaneous equations graphically
To solve simultaneous equations graphically you draw the straight lines and
then find their point of intersection. The coordinates of the point of intersection
give you the solutions x and y.
Note:
The calculator will only draw the graphs of functions that are expressed
explicitly. By that we mean functions that begin with ‘y =’ and have a function
that involves only x to the right of the equals sign. If the equations are written
in a different form, you will need to rearrange them before using your calculator
to solve them.
Solving simultaneous
equations using a nongraphical method is covered
in section 1.5.
Example 4
Solve the simultaneous equations 2x + y = 10 and x − y = 2 graphically with your GDC.
First rearrange both equations in the form y =
2x + y = 10
x−y=2
y = 10 − 2x
−y = 2 − x
y=x−2
To draw graphs y = 10 – 2x and y = x – 2.
Press Y= to display the Y= editor. The default graph type is Function,
so the form Y= is displayed.
Type 10 – 2x and press
ENTER
and x – 2 and press
ENTER .
Press ZOOM | 6:Z Standard to use the default axes which are
−10 ≤ x ≤ 10 and −10 ≤ y ≤ 10.
The calculator displays both straight line graphs
Y1 = 10 – 2x and
Y2 = x – 2
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Press
2nd CALC
| 5:intersect.
Press
ENTER .
Press
ENTER
to select the first curve.
Press
ENTER
to select the second curve.
Select a point close to the intersection using the
press ENTER .
and
keys and
The calculator displays the intersection of the two straight lines at the
point (4, 2).
The solutions are x = 4, y = 2.
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Simultaneous and quadratic equations
1.5 Solving simultaneous linear equations
When solving simultaneous equations in an examination, you do not need to
show any method of solution. You should simply write out the equations in the
correct form and then give the solutions. The calculator will do all the working
for you.
You will need to have the App
PlySmlt2 installed on your
GDC. This App is permitted
by IBO in your examination.
Example 5
Solve the equations:
2x + y = 10
x–y=2
Press APPS . You will see the dialog box as shown on the right. Choose the
App PlySmlt2 and press ENTER .
From the main menu, choose 2: SIMULT EQN SOLVER and press
ENTER .
The defaults are to solve two equations in two unknowns.
Note:
This is how you will use the linear equation solver in your
examinations. In your project, you might want to solve a more
complicated system with more equations and more variables.
Press
F5
and you will see the template on the right.
Type the coefficients from two equations into the
template, pressing ENTER after each number.
The equations must
be in the correct order.
Press F5 and the calculator will solve the equations, giving the solutions in
the as x1 and x2.
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The solutions are x = 4, y = 2.
Quadratic functions
1.6 Drawing a quadratic graph
Example 6
Draw the graph of y = x 2 − 2x + 3 and display it using suitable axes.
Press Y= to display the Y= editor. The default graph type is Function, so the
form Y= is displayed.
Type x 2 − 2x + 3 and press ENTER .
Press ZOOM | 6:Z Standard to use the default axes which are −10 ≤ x ≤ 10
and −10 ≤ y ≤ 10.
The calculator displays the curve with the default axes.
Adjust the window to make the quadratic curve fit the screen better.
1.7 Solving quadratic equations
When solving quadratic equations in an examination, you do not need to show
any method of solution. You should simply write out the equations in the
correct form and then give the solutions. The GDC will do all the
working for you.
Example 7
Solve 3x 2 − 4x − 2 = 0
Press APPS . You will see the dialog box as shown on the right. Choose the
App PlySmlt2 and press ENTER .
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From the main menu, choose 1: POLY ROOT FINDER and press
ENTER .
The defaults are to solve an equation of order 2 (a quadratic equation)
with real roots. You do not need to change anything.
Another dialog box opens for you to enter the equation.
The general form of the quadratic equation is a2x2 + a1x + a0 = 0,
so we enter the coefficients in a2, a1 and a0.
Here a2 = 3, a1 = −4 and a0 = −2. Be sure to use the (–) key to enter the
negative values.
Press ENTER after each value.
Press
F5
and the calculator will find the roots of the equation.
The solutions are x = −0.387 or x = 1.72 (3 sf).
1.8 Finding a local minimum or maximum point
Example 8
Find the minimum point on the graph of y = x 2 − 2x + 3.
Draw the graph of y = x 2 − 2x + 3 (See Example 19).
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Method 1 - using a table
You can look at the graph and a table of the values on the graph
by using a split screen.
Press
MODE
Press
GRAPH .
and select G-T.
The minimum value shown in the table is 2 when x = 1.
Look more closely at the values of the function around x = 1.
Change the settings in the table: Press 2nd TBLSET .
Set TblStart to 0.98
△Tbl to 0.01
Press 2nd TABLE to return to the graph and table screen.
Press
to move to the column containing y-values. This shows greater
precision in the box below the table.
The table shows that the function has larger values at points around (1, 2).
We can conclude that this is a local minimum on the curve.
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Method 2 – Using the minimum function
Press
2nd CALC
Press
ENTER .
| 3:minimum.
To find the minimum point you need to give the left and right bounds
of a region that includes it.
The calculator shows a point and asks you to set the left bound. Move
the point using the
and
keys to choose a position to the left of the
minimum.
Press
ENTER .
The calculator shows another point and asks you to set the right bound.
and
keys so that the region between the left
Move the point using the
and right bounds contains the minimum.
When the region contains the minimum press
Press ENTER again to supply a guess for
the value of the minimum.
The calculator displays the minimum
point on the curve at (1, 2).
ENTER .
In this example the value of x is not
exactly 1. This is due to the way the
calculator finds the point. You should
ignore small errors like this when you
write down the coordinates of the
point.
Example 9
Find the maximum point on the graph of y = −x 2 + 3x −4.
Press Y= to display the Y= editor. The default graph type is Function,
so the form Y= is displayed.
Type −x 2 + 3x −4 and press ENTER .
Press ZOOM | 6:Z Standard to use the default axes which are −10 ≤ x ≤ 10
and −10 ≤ y ≤ 10.
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The calculator displays the curve with the default axes.
Adjust the window to make the quadratic curve fit the screen better.
Method 1 - using a table
You can look at the graph and a table of the values on the graph by using a
split screen.
Press
MODE
and select G-T.
Press
GRAPH .
The maximum value shown in the table is –2 when x = 1 and x = 2.
Look more closely at the values of the function between x = 1 and x = 2.
Change the settings in the table: Press
2nd TBLSET .
Set TblStart to1.4
△Tbl to 0.01
Press
2nd
TABLE
to return to the graph and table screen.
Press
to move to the column containing y-values. This shows greater
precision in the box below the table.
Press
to scroll down until you find the maximum value of y.
The table shows that the function has smaller values at points around
(1.5, –1.75). We can conclude that this is a local maximum on the curve.
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Method 2 – Using the maximum function
Press
2nd CALC
Press
ENTER .
| 4:maximum.
To find the maximum point you need to give the left and right bounds of a
region that includes it.
The calculator shows a point and asks you to set the left bound. Move
the point using the
and
keys to choose a position to the left of the
maximum.
Press
ENTER .
The calculator shows another point and asks you to set the right bound.
and
keys so that the region between the left
Move the point using the
and right bounds contains the minimum.
When the region contains the minimum press
Press ENTER again to supply a guess
for the value of the minimum.
The calculator displays the
maximum point on the curve
at (1.5, –1.75).
ENTER .
In this example the value of x is not
exactly 1.5. This is due to the way the
calculator finds the point. You should
ignore small errors like this when you
write down its coordinates.
Exponential functions
1.9 Drawing an exponential graph
Example 10
Draw the graph of y = 3x + 2.
Press Y= to display the Y= editor. The default graph type is Function, so
the form Y= is displayed.
Type 3x + 2 and press
ENTER .
>
X, T, H, n
(Note: Type 3
from the exponent.)
to enter 3x. The
returns you to the baseline
Press ZOOM | 6:ZStandard to use the default axes which are −10 ≤ x ≤ 10
and −10 ≤ y ≤ 10.
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The calculator displays the curve with the default axes.
Adjust the window to make the exponential curve fit the screen better.
1.10 Finding a horizontal asymptote
Example 11
Find the horizontal asymptote to the graph of y = 3x + 2.
Draw the graph of y = 3x + 2 (see Example 22).
You can look at the graph and a table of the values on the graph by using a
split screen.
Press
MODE
and select G-T.
Press
GRAPH .
The values of the function are clearly decreasing as x → 0.
Press
Press
2nd
TABLE
to switch to the table.
to scroll up the table.
The table shows that as the values of x get smaller, Y1 approaches 2.
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TI-84 Plus
Eventually the value of Y1 displayed in the table reaches 2.
to move to the column containing y-values. This shows greater
Press
precision in the box below the table. You can see, at the bottom of the
screen, that the actual value of Y1 is 2.00000188168...
We can say that Y1 ՜ 2 as x ՜ −∞.
The line x = 2 is a horizontal asymptote to the curve y = 3x + 2.
Logarithmic functions
1.11 Evaluating logarithms
Example 12
Evaluate log10 3.95, ln10.2 and log5 2.
Press ALPHA F2 | 5:logBASE(to open the log template.
Enter the base and the argument then press ENTER .
For natural logarithms it is possible to use the same method, with the base
equal to e, but it is quicker to press LN .
Note that the GDC will evaluate logarithms with any base without having
to use the change of base formula.
1.12 Finding an inverse function
The inverse of a function can be found by interchanging the x and y values.
Geometrically this can be done by reflecting points in the line y = x.
Example 13
Show that the inverse of the function y = 10 x is y = log10 x by reflecting y = 10 x in the line y = x.
Draw the line y = x so that it can be recognised as the axis of reflection.
Press Y= to display the Y= editor. The default
graph type is Function, so the form Y= is displayed.
Type x and press ENTER .
Note: Type 1 0 ^ X, T, H, n
Type 10 x and press ENTER .
x
to enter 10 . The
returns you
to the baseline from the exponent.
Press WINDOW and choose options as shown.
This will set up square axes −4.7 ≤ x ≤ 4.7 and −3.1 ≤ y ≤ 3.1. with the
same horizontal and vertical scales.
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Press GRAPH .
The graphs of y = x and y = 10x are displayed.
Press 2nd DRAW | 8:DrawInv.
Then press ALPHA F4 and choose Y2.
Press ENTER .
Alternatively press LOG X, T, H, n
to enter log(x). LOG is a shorter
way to enter log10.
Press GRAPH .
The graphs are displayed.
The calculator will display the inverse of the function y = 10 x.
Press Y = to display the Y= editor.
Type log(x).
Press LOG X, T, H, n to enter
log(x). LOG is a shorter way
to enter log10.
Press GRAPH to display the graphs of y = x, y = 10 x and y = log10 x.
The inverse function and the logarithmic function coincide, showing that
y = log10 x is inverse of the function y = 10 x.
1.13 Drawing a logarithmic graph
Example 14
Draw the graph of y = 2log10 x + 3.
Press Y= to display the Y= editor. The default graph type is Function, so
the form Y= is displayed.
Press ALPHA F2 | 5:logBASE( to open the log template.
Enter the base and the argument then press ENTER .
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Type 2log10 (x) + 3 and press ENTER .
Press ZOOM 6:XStandard so that the calculator displays the curve with the
default axes.
The calculator displays the curve with the default axes.
Change the axes to make the logarithmic curve fit the screen better.
Trigonometric functions
1.14 Degrees and radians
Work in trigonometry will be carried out either in degrees or radians. It is
important, therefore, to be able to check which mode the calculator is in and to
be able to switch back and forth
Example 15
Change angle settings from radians to degrees and from degrees to radians.
Press MODE .
Select either RADIAN or DEGREE using the
Press ENTER .
Press
2nd
keys.
QUIT .
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1.15 Drawing a trigonometric graph
Example 16
Draw the graph of y
S·
§
2sin ¨ x ¸ 1.
6
©
¹
Press Y= to display the Y= editor. The default graph type is Function, so
the form Y= is displayed.
S
Type y 2sin §¨ x ·¸ 1 and press ENTER .
6¹
©
Press ZOOM 7:ZTrig.
The default axes are −6.15 ≤ x ≤ 6.15 and −4 ≤ y ≤ 4.
The notation sin2x, cos2x, tan2x, …
is a mathematical convention that
has little algebraic meaning. To enter
these functions on the GDC, you
should enter (sin(x))2, etc. However,
the calculator will conveniently
interpret sin(x)2 as (sin(x))2.
More complicated functions
Follow the same GDC procedure
when solving simultaneous equations
graphically and solving a combined
quadratic and exponential equation.
See Examples 18 and 24.
1.16 Solving a combined quadratic
and exponential equation
Example 17
Solve the equation x 2 − 2x + 3 = 3 · 2−x + 4
To solve the equation, find the point
of intersection between the quadratic
function y1 = x2 − 2x + 3 and the
exponential function y2 = 3 × 2−x + 3.
Press Y= to display the Y= editor. The default graph type is Function, so
the form Y= is displayed.
Type x2 − 2x + 3 in Y1 and press ENTER . Then type 3 × 2−x + 4 in Y2
and press ENTER .
>
(Note: Type 2
(–) X, T, H, n
baseline from the exponent.)
to enter 2–x. The
returns you to the
Press ZOOM | 6:Z Standard to use the default axes which are −10 ≤ x ≤ 10
and −10 ≤ y ≤ 10.
The calculator displays the curves with the default axes.
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Adjust the window to make the quadratic curve fit the screen better.
Press
2nd CALC
| 5:intersect.
Press
ENTER .
Press
ENTER
to select the first curve.
Press
ENTER
to select the second curve.
Select a point close to the intersection using the
and press ENTER .
and
keys
The calculator displays the intersection of the two straight lines at the point
(2.58, 4.50).
The solutions are x = 2.58 and y = 4.50.
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1.17 Using sinusoidal regression
Example 18
It is known that the following data can be modelled using a sine curve:
x
0
1
2
3
4
5
6
7
y
6.9
9.4
7.9
6.7
9.2
8.3
6.5
8.9
Use sine regression to find a function to model this data.
Press STAT |1:Edit and press F3 .
Type the x-values in the first column (L1) and the y-values in the second
column (L2).
Press ENTER or
after each number to move down to the next cell.
Press
to move to the next column.
You can use columns from L1 to L6 to enter the lists.
Press 2nd STAT PLOT and eto select Plot1.
Select On, choose the scatter diagram option, XList as L1 and Ylist as L2.
You can choose one of the three types of mark.
Press ZOOM 9:ZoomStat.
Adjust your window settings to show your data and the x- and y-axes.
You now have a scatter plot of x against y.
Press
2nd
î to return to the Home screen.
Press
STAT
CALC | C:SinReg.
Press
2nd
Press
F3
L1
,
2nd
æ,
ALPHA
F4
choose Y1 and press
F3
.
again.
On screen, you will see the result of the sinusoidal regression.
The equation is in the form y = asin(bx + c) + d and you will see the values
of a, b, c and d displayed separately.
The equation of the sinusoidal regression line is
y = 1.51sin(2.00x − 0.80) + 7.99
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Press
GRAPH
to return to the Graphs page.
Press .
The regression line is now shown in Y1. You can see the full equation if you
scroll to the right.
2 Differential calculus
Finding gradients, tangents and maximum
and minimum points
2.1 Finding the gradient at a point
Example 19
Find the gradient of the cubic function y = x3 − 2x2 − 6x + 5 at the point where x = 1.5.
Press Y= to display the Y= editor. The default graph type is Function, so the
form Y= is displayed.
Type y = x3 − 2x2 − 6x + 5 and press
>
(Note: Type X, T, H, n
from the exponent.)
3
ENTER .
to enter x 3. The
returns you to the baseline
Press ZOOM | 6:ZStandard to use the default axes which are −10 ≤ x ≤ 10 and
−10 ≤ y ≤ 10.
Adjust the window to make the cubic curve fit the screen better.
Press
2nd CALC
Press
ENTER
Press
1
.
| 6: dy/dx.
.
5
ENTER
.
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The calculator displays the gradient
of the curve at the point where x = 1.5.
The gradient is –5.25.
In this example the value of xdy/dx
is not exactly –5.25. This is due to
the way the calculator finds the point
gradient. You should ignore small
errors like this when you write down
the coordinates of a gradient at a the
point.
2.2 Drawing a tangent to a curve
Example 20
Draw a tangent to the curve y = x 3 − 2x2 − 6x + 5 where x = – 0.5.
First draw the graph of y = x 3 − 2x2 − 6x + 5 (see Example 30).
Press
2nd
DRAW
.
Choose 5:Tangent.
Press
ENTER .
Press (–)
0
.
5
ENTER .
The equation of the tangent is
y = −3.25x + 5.75
In this example the values −3.25 and
5.75 are not shown as being exact.
This is due to the way the calculator
finds the values. You should ignore
small errors like this when you write
down the equation of a tangent.
2.3 Finding maximum and minimum points
Example 21
Find the local maximum and local minimum points on the cubic curve.
First draw the graph of y = x 3 − 2x2 − 6x + 5 (see Example 30).
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Press
2nd CALC
Press
ENTER .
| 3:minimum.
To find the minimum point you need to give the left and right bounds of a
region that includes it.
The calculator shows a point and asks you to set the left bound. Move
the point using the
and
keys to choose a position to the left of the
minimum.
Press
ENTER .
The calculator shows another point and asks you to set the right bound.
and
keys so that the region between the left
Move the point using the
and right bounds contains the minimum.
When the region contains the minimum press
Press
ENTER
ENTER .
again to supply a guess for the value of the minimum.
The calculator displays the local minimum at the point (2.23, −7.24).
Press 2nd CALC | 3:maximum ENTER . To find the local maximum point
on the curve in exactly the same way.
The maximum point is (−0.897, 8.05).
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2.4 Finding a numerical derivative
Using the calculator it is possible to find the numerical value of any derivative
for any value of x. The calculator will not, however, differentiate a function
algebraically. This is equivalent to finding the gradient at a point graphically.
Example 22
If y =
x +3
dy
, evaluate
x
dx
x =2
Press ALPHA F2 .
Choose 3: nDeriv( to choose the derivative template.
Enter x and the function in the template. Enter the value 2.
Press ENTER .
x +3
The calculator shows that the value of the first derivative of y =
x
is −0.75 when x = 2.
2.5 Graphing a numerical derivative
Although the calculator can only evaluate a numerical derivative at a point, it
will graph the gradient function for all values of x.
Example 23
If y =
x
dy
.
, draw the graph of
x +3
dx
Press Y= to display the Y= editor. The default graph type is Function,
so the form Y= is displayed.
Press ALPHA F2 .
Choose 3: nDeriv (to choose the derivative template.
{ Continued on next page
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TI-84 Plus
x
In the template enter x, the function
and the value x.
x +3
Press ENTER .
Press ZOOM 6:ZStandard.
The calculator displays the graph of the numerical derivative function
x
.
of y =
x +3
Example 24
Find the values of x on the curve y =
x3
+ x 2 − 5x + 1 where the gradient is 3.
3
Press Y= to display the Y= editor. The default graph type is Function,
so the form Y= is displayed.
Press ALPHA F2 .
Choose 3: nDeriv( to choose the derivative template.
x3
+ x 2 − 5x + 1 and the value x.
In the template enter x, the function
3
Press ENTER .
Press ZOOM 6:ZStandard.
The calculator displays the graph of the numerical derivative function
of y =
x3
+ x 2 − 5 x + 1.
3
{ Continued on next page
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TI-84 Plus
Press Y= to display the Y= editor.
Enter the function Y2 = 3.
Press GRAPH .
The calculator now displays the curve and the line y = 3.
To find the points of intersection between the curve and the line.
Press 2nd CALC | 5:intersect.
Press
ENTER .
Press
ENTER
to select the first curve.
Press
ENTER
to select the second curve.
Select a point close to the intersection using the
and press ENTER .
Repeat for the second point of intersection.
and
keys
The curve has gradient 3 when x = –4 and x = 2
In this example the value of x is not
exactly 2. This is due to the way the
calculator finds the point. You should
ignore small errors like this when
you write down the coordinates of a
gradient at a point.
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TI-84 Plus
3 Integral calculus
The calculator can find the values of definite integrals either on a calculator
page or graphically. The calculator method is quicker, but the graphical method
is clearer and shows discontinuities, negative areas and other anomalies that
can arise.
3.1 Finding the value of an indefinite integral
Example 25
8
⌠ 
Evaluate   x −
⌡2 
3 
 dx .
x
Press ALPHA F2 .
Choose 4: fnlnt( to choose the integral template.
In this example you will also use templates to enter the rational function
and the square root.
Enter the upper and lower limits, the function and x in the template.
Press ENTER .
The value of the integral is 21.5 (to 3 sf).
3.2 Finding the area under a curve
Example 26
Find the area bounded by the curve y = 3x 2 − 5, the x-axis and the lines x = −1 and x = 1.
Press Y= to display the Y= editor. The default graph type is Function, so
the form Y= is displayed.
Type y = 3x 2 − 5 and press ENTER .
{ Continued on next page
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TI-84 Plus
Press ZOOM 6:ZStandard.
The default axes are −10 ≤ x ≤ 10 and −10 ≤ y ≤ 10.
Adjust the window settings to view the curve better.
Press
2nd CALC
7:∫f(x)dx.
The calculator prompts you to enter the lower limit for the integral.
Type –1 and press ENTER .
Be sure to use the (–) key.
Type 1 and press
ENTER .
The area found is shaded and the value of the integral (–8) is shown on the
screen.
Note: since the area lies below the x-axis in this case, the integral is
negative.
The required area is 8.
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4 Vectors
Scalar product
4.1 Calculating a scalar product
Example 27
There is no scalar product
function on the TI-84 plus,
but you can find the result
by multiplying the vectors
as lists and then finding the
sum of the terms in the list.
Evaluate the scalar products:
a
 1   −3 
 ⋅ 
3  4 
a
Press
2nd
 1  3 
   
b  −1 ⋅  2 
 4   −1
   
LIST | MATH | 5:sum(.
Enter the vectors as lists using curly brackets { }. Separate the terms of the
vectors using commas.
Multiply the two vector lists together.
Close the bracket and press
ENTER .
 1   −3 
 ⋅ =9
3  4 
b
Press
2nd
LIST | MATH | 5:sum(.
{ Continued on next page
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TI-84 Plus
Enter the vectors as lists using curly brackets { }. Separate the terms of the
vectors using commas.
Multiply the two vector lists together.
Close the bracket and press
ENTER .
 1  3 
   
 −1 ⋅  2  = − 3
 4   −1
   
4.2 Calculating the angle between two vectors
G
G
The angle θ between two vectors a and b , can be calculated using the formula
G G
a•b
q = ar cos  G G 
a b
Example 28
G
G
G G
Calculate the angle between 2 i + 3 j and 3 i − j
Press MODE .
Select either RADIAN or DEGREE (according to the units you need your
answer in) using the
keys.
Press ENTER .
Press
2nd
QUIT .
Press
2nd
DISTR
Press
ALPHA
F1
.
and select the fraction template 1:n/d
{ Continued on next page
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TI-84 Plus
Press
2nd
LIST | MATH | 5:sum(.
Enter the vectors as lists using curly brackets { }. Separate the terms of the
vectors using commas.
Multiply the two vector lists together.
To calculate the magnitudes of the vectors use the formula
G G
a i + b j = a2 + b 2
Use the
Press
key to exit the templates before entering the final bracket.
ENTER .
G
G
G G
The angle between 2 i + 3 j and 3 i − j is 74.7°.
5 Statistics and probability
You can use your GDC to draw charts to represent data and to calculate basic
statistics such as mean, median, etc. Before you do this you need to enter
the data in a list.
Entering data
There are two ways of entering data: as a list or as a frequency table.
5.1 Entering lists of data
Example 29
Enter the data in the list: 1, 1, 3, 9, 2.
Press STAT 1: Edit and press ENTER .
Type the numbers in the first column (L1).
Press ENTER or after each number to move
down to the next cell.
L1 will be used later when you want to
make a chart or to do some calculations
with this data. You can use columns from
L1 to L6 to enter the list.
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TI-84 Plus
5.2 Entering data from a frequency table
Example 30
Enter the data in the table:
Number
1
2
3
4
5
Frequency
3
4
6
5
2
Press STAT |1:Edit and press ENTER .
Type the numbers in the first column (L1) and
the frequencies in the second column (L2).
Press ENTER or after each number to move
down to the next cell.
Press to move to the next column.
L1 and L2 will be used later when you want
to make a chart or to do some calculations
with this data. You can use columns from L1
to L6 to enter the lists.
Drawing charts
Charts can be drawn from a list or from a frequency table.
5.3 Drawing a frequency histogram from a list
Example 31
Draw a frequency histogram for this data: 1, 1, 3, 9, 2.
Enter the data in L1 (see Example 5).
Press 2nd STAT PLOT and ENTER to select Plot1.
Select On, choose the histogram option and
leave XList as L1 and Freq as 1.
Press ZOOM | 9:Stat.
You may need to
The automatic scales
do not usually give the delete any function
graphs. Y=
best display of the
histogram. You will
need to change the default values.
Press WINDOW and choose options as shown.
Xmin and Xmax should include the range
of the data.
Ymin and Ymax should include the
maximum frequency and should go below
zero.
Xscl will define the width of the bars.
Press
TRACE .
key to move to each of the bars
Use the
and display their value and frequency.
You should now see a frequency histogram
for the data in the list 1, 1, 3, 9, 2.
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TI-84 Plus
5.4 Drawing a frequency histogram from a frequency table
Example 32
Draw a frequency histogram for this data: Number
Frequency
1
2
3
4
5
3
4
6
5
2
Enter the data in L1 and L2 (see Example 6).
Press 2nd STAT PLOT and ENTER to select Plot 1.
Select On, choose the histogram option
and leave XList as L1 and Freq as L2.
Press ZOOM | 9:Stat.
The automatic scales
You may need to
do not usually give the delete any function
best display of the
graphs. Y=
histogram. You will
need to change the default values.
Press WINDOW and choose options as shown.
Xmin and Xmax should include the range
of the data.
Ymin and Ymax should include the
maximum frequency and should go below
zero.
Xscl will define the width of the bars.
Press TRACE .
Use the
key to move to each of the bars
and display their value and frequency.
You should now see a frequency histogram
for the data in the list 1, 1, 3, 9, 2.
5.5 Drawing a box and whisker diagram from a list
Example 33
Draw a box and whisker diagram for this data:
1, 1, 3, 9, 2.
Enter the data in L1 (see Example 5).
Press 2nd STAT PLOT and ENTER to select Plot 1.
Select On, choose the box and whisker
option and leave XList as L1 and Freq
as 1.
{ Continued on next page
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TI-84 Plus
Press ZOOM | 9:Stat.
The automatic scales do
not usually give the best
display of the box and
whisker diagram. You
will need to change the
default values.
Press
WINDOW
You may need
to delete any
function graphs.
Y=
and choose options as shown.
Xmin and Xmax should include the range
of the data.
Ymin and Ymax do not affect the way in
which the diagram is displayed.
Press TRACE .
Use the
key to move the cursor over the
plot to see the quartiles, Q1 and Q3, the
median and the maximum and minimum
values.
5.6 Drawing a box and whisker diagram
from a frequency table
Example 34
Draw a box and whisker diagram for this data:
Number
1
2
3
4
5
Frequency
3
4
6
5
2
Enter the data in L1 and L2
(see Example 6).
Press 2nd STAT PLOT and ENTER to select Plot 1.
Select On, choose the box and whisker
diagram option and leave XList as L1 and
Freq as L2.
Press ZOOM | 9:Stat.
The automatic scales do
not usually give the best
display of the box and
whisker diagram. You
will need to change the
default values.
You may need
to delete any
function
graphs. Y=
{ Continued on next page
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TI-84 Plus
Press WINDOW and choose options as shown.
Xmin and Xmax should include the range
of the data.
Ymin and Ymax do not affect the way in
which the diagram is displayed.
Press TRACE .
Use the
key to move the cursor over the
plot to see the quartiles, Q1 and Q3, the
median and the maximum and minimum
values.
Calculating statistics
You can calculate statistics such as mean, median, etc. from a list,
or from a frequency table.
5.7 Calculating statistics from a list
Example 35
Calculate the summary statistics for this data: 1, 1, 3, 9, 2
Enter the data in L1 (see Example 5).
Press
STAT
Type
2nd
| CALC | 1:1-Var Stats.
L1
and press
ENTER .
The information shown will not fit on a single
screen. You can scroll up and down to see it all.
The statistics calculated for the data are:
mean
sum
sum of squares
sample standard deviation
population standard
deviation
number
minimum value
lower quartile
median
upper quartile
maximum value
x
∑x
∑x
2
sx
σx
n
MinX
Q1
Med
Q3
MaxX
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TI-84 Plus
5.8 Calculating statistics from a frequency table
Example 36
Calculate the summary statistics for this data:
Number
1
2
3
4
5
Frequency
3
4
6
5
2
Enter the data in L1 and L2 (see Example 6).
Press STAT | CALC | 1:1-Var Stats.
Type
2nd
,
L1
2nd
L2
and press
ENTER .
The information shown will not fit on a single
screen. You can scroll up and down to see it all.
The statistics calculated for the data are:
mean
x
sum
∑x
sum of squares
∑x
sample standard deviation
sx
population standard
deviation
σx
number
minimum value
lower quartile
median
upper quartile
maximum value
2
n
minX
Q1
Med
Q3
MaxX
5.9 Calculating the interquartile range
Example 37
Calculate interquartile range for this data:
Number
1
2
3
4
5
Frequency
3
4
6
5
2
The interquartile range is the
difference between the upper and
lower quartiles (Q3− Q1).
First calculate the summary statistics for this data (see Example 12).
(Note: The values of the summary statistics are stored after One-Variable
Statistics have been calculated and remain stored until the next time they
are calculated.)
Press VARS | 5:Statistics... | PTS | 9:Q3 ENTER – VARS | 5:Statistics... |
PTS | 7:Q1 ENTER .
The calculator now displays the result:
Interquartile range = Q3 – Q1 = 2
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TI-84 Plus
5.10 Using statistics
The calculator stores the values you calculate in One-Variable Statistics so that
you can access them in other calculations. These values are stored until you
do another One-Variable Statistics calculation.
Example 38
Calculate the x + σ x for this data:
Number
1
2
3
4
5
Frequency
3
4
6
5
2
First calculate the summary statistics for this data (see Example 12).
(Note: The values of the summary statistics are stored after One-Variable
Statistics have been calculated and remain stored until the next time they
are calculated.)
Press VARS | 5:Statistics... | 2: x ENTER – VARS | 5:Statistics... 4:σ x ENTER .
The calculator now displays the result:
x + σ x = 4.15 (to 3 sf)
Calculating binomial probabilities
5.11 The use of nCr
Example 39
8
Find the value of   (or 8C3 ).
3
Press
Press
MATH
Press
3
Press
ENTER .
8
.
3:nCr.
ENTER .
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TI-84 Plus
Example 40
4
List the values of   for r = 0, 1, 2, 3, 4.
r 
Press Y= to display the Y= editor. The default graph type is Function, so
the form Y= is displayed.
Press 4 .
Press m 3:nCr.
Press X, T, H, n ENTER .
Press 2nd TABLE .
The table shows that
4
4
4
4
  = 1,   = 4,   = 6,   = 4
0
1
2
3
4
and   = 1
4
You may need to reset the
start value and incremental
values for the table using
2nd TBLSET
5.12 Calculating binomial probabilities
Example 41
X is a discrete random variable and X ~ Bin(9, 0.75).
Calculate P(X = 5)
9
P ( X = 5 ) =   0.755 0.254
5
The calculator can find this value directly.
Press
2nd
DISTR
A:binompdf(.
Enter 9 as trials, 0.75 as p and 5 as x.
Select Paste and press ENTER
Press ENTER again
You should enter
the values: n (numtrials), p
and x, in order.
The calculator shows that P (X = 5) = 0.117 (to 3 sf).
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TI-84 Plus
Example 42
X is a discrete random variable and X ~ Bin(7, 0.3).
Calculate the probabilities that X takes the values {0, 1, 2, 3, 4, 5, 6, 7}.
Press 2nd DISTR A:binompdf(.
Enter 7 as trials, 0.3 as p and leave x blank.
Select Paste and press ENTER
Press ENTER again
You should enter
the values: n (numtrials), p
and x, in order.
The calculator displays each of the probabilities.
To see the remaining values scroll the screen to the right.
The list can also be transferred as a list.
Press
STO
Press
ENTER .
2nd
L1
.
Press STAT 1:Edit…
The binomial probabilities are now displayed in the first column.
Example 43
X is a discrete random variable and X ~ Bin(20, 0.45).
Calculate
a the probability that X is less than or equal to 10.
b the probability that X lies between 5 and 15 inclusive.
c the probability that X is greater than 11.
Press
2nd
DISTR
B:binomcdf(.
You are given the lower
bound probability so you
have to calculate other
probabilities using this.
You should enter
the values: n (numtrials),
p and x, in order.
{ Continued on next page
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TI-84 Plus
a
Enter 30 as trials, 0.45 as p and 10 as x.
Select Paste and press ENTER
Press ENTER again
P(X ≤ 10) = 0.751 (to 3 sf).
b
P(5 ≤ X ≤ 15) = P(X ≤ 15) − P(X ≤ 4)
Press
2nd
DISTR
B:binomcdf(
Enter 20 as trials, 0.45 as p and 10 as x.
Select Paste and press ENTER
Type (–) and then Press 2nd DISTR B:binomcdf(
Enter 20 as trials, 0.45 as p and 4 as x.
Select Paste and press ENTER
Press ENTER again
P(5 ≤ X ≤ 15) = 0.980 (to 3 sf).
c P(X > 11) = 1 − P(X ≤ 11)
Enter 1 – and then Press 2nd
DISTR
B:binomcdf(
Select Paste and press ENTER
Press ENTER again
P(X > 11) = 0.131 (to 3 sf).
Calculating normal probabilities
5.13 Calculating normal probabilities from X-values
Example 44
A random variable X is normally distributed with a mean of 195 and a standard deviation
of 20 or X ∼ N(195, 202). Calculate
a the probability that X is less than 190.
b the probability that X is greater than 194.
c the probability that X lies between 187 and 196.
Press
2nd
Press
ENTER .
a
DISTR
| 2:normalcdf(.
You should enter the
values, Lower Bound,
Upper Bound, μ and σ,
in order.
The value E99 is the largest value that
can be entered in the GDC and is used
in the place of λ. It stands for 1 × 1099
(–E99 is the smallest value and is
used in the place of –λሻǤ To enter the
E, you need to press 2nd
EE .
P (X < 190)
Enter Lower Bound as െE99, Upper Bound as 190, µ to 195 and σ to 20.
P(X < 190) = 0.401 (to 3 sf)
{ Continued on next page
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TI-84 Plus
b
P (X < 194)
Enter Lower Bound as 194, Upper Bound as E99, µ as 195 and σ as 20.
P(X > 194) = 0.520 (to 3 sf)
c
P (187 < X < 196)
Enter Lower Bound as 187, Upper Bound as 196, µ as 195 and σ as 20.
P(187 < X < 196) = 0.175 (to 3 sf)
5.14 Calculating X-values from normal probabilities
In some problems you are given probabilities and have to calculate the associated
values of X. To do this, use the invNorm function.
When using the Inverse Normal function,
make sure you find the probability on the
correct side of the normal curve. The areas
are always the lower tail, that is they are
always in the form P (X < x) (see Example 26).
If you are given the upper tail P (X > x), you
must first subtract the probability from 1
before you can use invNorm (see example 27).
Example 45
A random variable X is normally distributed with a mean of 75
and a standard deviation of 12 or X ∼ N(75, 122).
If P (X < x) = 0.4, find the value of x.
Press
2nd
Press
ENTER .
DISTR
| 3:invNorm(.
You are given a lower-tail probability
so you can find P (X < x) directly.
You should enter the values: area
(probability), μ and σ, in order.
Enter area (probability) as 0.4, μ as 75 and σ as 12.
So if P (X < x) = 0.4 then x = 72.0 (to 3 sf).
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Example 46
A random variable X is normally distributed with a mean
of 75 and a standard deviation of 12 or X ∼ N(75, 122).
If P (X > x) = 0.2, find the value of x.
Press
2nd
Press
ENTER .
DISTR
| 3:invNorm(.
You are given an upper-tail probability so
you must first find P (X < x) = 1 − 0.2 = 0.8.
You can now use the invNorm function as
before.
You should enter
the values: area
(probability), μ
and σ, in order.
Enter area (probability) as 0.8, μ as 75 and σ as 12.
So if P (X > x) = 0.2 then x = 85.1 (to 3 sf).
85.1
0.8
40
60
0.2
80
80
100
x
[ This sketch of a normal
distribution curve shows this
value and the probabilities
from Example 29.
Scatter diagrams, linear regression
and the correlation coefficient
5.15 Scatter diagrams
Example 47
Consider this data that is approximately connected by a linear function.
x
y
1.0
4.0
2.1
5.6
2.4
9.8
3.7
10.6
5.0
14.7
Find the equation of the least squares regression line of y on x.
b Find Pearson’s product-moment correlation coefficient.
c Use the equation to predict the value of y when x = 3.0.
a
Press
STAT
|1:Edit and press
ENTER .
Type the values of x in the first column (L1)
and the values of y in the second column (L2).
Press
Press
or
after each number to move down to the next cell.
to move to the next column.
ENTER
You can use columns from L1 to L6 to enter the lists.
{ Continued on next page
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TI-84 Plus
Press
2nd STAT PLOT
and
ENTER
to select Plot1.
Select On, choose the scatter diagram option, XList as L1 and Ylist as L2.
You can choose one of the three types of mark.
Press ZOOM | 9:Stat.
The automatic scales do not usually give the best
display of the scatter diagram. You will need to
change the default values.
Press
WINDOW
and choose options as shown.
Xmin and Xmax should include the range of
the x-data.
Ymin and Ymax should include the range of
the y-data.
You may need to
delete any function
graphs. Y=
You need to include
zero in the range if you
want to show the axes
on the graph.
You now have a scatter graph of y against x.
{ Continued on next page
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In order to see the Pearson’s product-moment correlation coefficient, you
need to have diagnostics on your GDC switched on.
Press MODE and use
to scroll down to the second screen. Set STAT
DIAGNOSTICS to ON and press ENTER .
Then press 2nd QUIT to return to the home screen.
Press
STAT
| CALC |4:LinReg(ax + b).
Press
2nd
Press
ALPHA
Press
ENTER
L1
F4
,
2nd
L2
and press
,.
ENTER
to select Y1.
again.
You will see the coefficients of the equation of the
least squares regression line and the value r of the
correlation coefficient.
The equation is y = 2.63x + 1.48 (to 3 sf).
The value of r is 0.955 (to 3 sf).
Press GRAPH and you will see the least squares regression line and the data
points that you plotted previously.
{ Continued on next page
© Oxford University Press 2012: this may be reproduced for class use solely for the purchaser’s institute
Using a graphic display calculator
43
TI-84 Plus
Press
TRACE
and use the
keys to move the trace along the line.
The cursor moves between the data points.
Press
to move onto the line itself.
It is not possible to move the trace point to an exact value, so get as close
to x = 3 as you can.
From the graph, you have found that y is approximately 9.5 when x = 3.0.
Press 3
ENTER .
The cursor now moves to exactly 3.0.
When x = 3.0, an estimate of the value of y is 9.36, from the graph.
© Oxford University Press 2012: this may be reproduced for class use solely for the purchaser’s institute
Using a graphic display calculator
44